Constraints and Metrics in Technical NetworksGoals of This ClassDesigned and Grown SystemsPearson Coefficient for Technical SystemsPearson Coefficient for Canonical SystemsClosed Form ResultsNested Self-Similar NetworksTree with Diminishing Branching RatioTrees with Branching Ratio bAlignment of System Type and Canonical ShapeNetwork Models of Technical SystemsDesign of Distribution SystemsSpatial Distribution Networks -1Spatial distribution networks (Magee slides)Spatial distribution networks -2Spatial distribution networks - 3Spatial distribution networks -4Spatial distribution networks 5London UndergroundTokyo JR East Linesand SubwaysTokyo SubwaysMoscow MetroMoscow Regional RailMoscow Metro and Regional RailBerlin U-bahn and S-bahnMunich U-bahn and S-bahnParis Metro and RERCar and Train Traffic, ParisFood WebsCanonical Forms in Food WebsUK GrasslandChesapeake BayBroom (Scotch Broom Grass)Little Rock Lake, WIr for Food Webs: Toy Example Seeking the Reason for ± rTrophic Species Affect rMechanical AssembliesAverage Nodal Index for Mechanical Assemblies ≈ 1.5*2Average Nodal Index Does Not Grow with Network Size, Unlike Most Networks: Why?Constraint as a Limit on ConnectivityConstraint and <k>What Matters in Assembly NetworksExample Assembly: Exercise WalkerNetwork Diagram of WalkerWalker Horizontal Functional LoopsWalker Vertical Functional LoopsV-8 Engine Functional LoopsSummary Properties of Several Big Networks (Newman)Additional Networks8/24/2006 © Daniel E Whitney 1997-2006 1/49Technical networksConstraints and Metrics in Technical Networks8/24/2006 © Daniel E Whitney 1997-2006 2/49Technical networksGoals of This Class• Metrics as indicators of system properties that may be related to structure or behavior• Their relation to the constraints under which the systems evolved or in view of which the systems were designed• Modeling problems8/24/2006 © Daniel E Whitney 1997-2006 3/49Technical networksDesigned and Grown Systems• Designed implies some degree of top-down control of the architecture• Grown does not mean random• Social systems– Designed: organizations, supply chains– Grown: coauthors, company directors• Technical systems– Designed: assemblies, PSTN, factories, national highway network– Grown: regional or national electric grid, local roads outside of Northwest Territory• Harder to classify (social?, grown?)– A city and its water supply or subway system8/24/2006 © Daniel E Whitney 1997-2006 4/49Technical networksPearson Coefficient for Technical Systems• A widely studied metric that captures some elements of structure and possibly is related to behavior• Distribution systems - grids or stars or trees or trees with cross-links• Mechanical assemblies - trees• Electric and electronic circuits - should be grids– Computer motherboards have a few nodes with huge k– What do they look like inside?– Coarse-graining8/24/2006 © Daniel E Whitney 1997-2006 5/49Technical networksPearson Coefficient for Canonical Systems• Trees, cross-linked trees, and stars have r < 0• Balanced binary trees have r = -1/3• Trees with diminishing branching ratio have r > 0• Trees with big branching ratios explode and r approaches -1• Finite grids have r approaching 2/3• Clusters with pendants at each cluster node have r < 08/24/2006 © Daniel E Whitney 1997-2006 6/49Technical networksClosed Form Results1234512345Property Pure Binary Tree Binary Tree with Cross-linking ksum 2n+1− 4 3*2n−10 ksqsum 10*2n−1−14 13*2n− 64 x → 2.5 as n becomes large (>~6) →133 as n becomes large (>~6) Pearson numerator ~2n(3−x )(1−x )+(ksum−2n)(3−x )2 ~2n(5− x )(1−x )+(ksum−2n)(5−x )2 Pearson denominator ~2n−1(1−x )2+(ksum−2n−1)(3−x )2 ~2n−1(1− x )2+(ksum−2n−1)(5−x )2 r →−13 as n becomes large →−15 as n becomes large l r =16(2−x )(3− x ) + 8(l − 3)(3 − x )22(2 − x )2+12(l − 2)(3− x )2→23Note: Western Power Grid r = 0.0035Bounded grid8/24/2006 © Daniel E Whitney 1997-2006 7/49Technical networksNested Self-Similar Networksnestedr = - 0.25, c = 0.625nested2r=-0.0925,c=0.5500Probably, r = 0in the limit as thenetwork grows8/24/2006 © Daniel E Whitney 1997-2006 8/49Technical networksTree with Diminishing Branching Ratio16 times8times4times1nodewithk=1616 nodes with k = 98*16 = 128 nodes with k = 54*8*16 = 512 nodes with k = 32*4*8*16 = 1024 nodes with k = 12timesr = 0.381668/24/2006 © Daniel E Whitney 1997-2006 9/49Technical networksTrees with Branching Ratio bUsing approximate formula; tested in matlab with tree-generatorwritten by Mo-Han Hsieh. Actual values are a bit more negative than given by approximate formula for b > 2.Pearson Coeff r for Branching Trees-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100 20 40 60 80 100 120Branching Ratiorb=2b=3b=48/24/2006 © Daniel E Whitney 1997-2006 10/49Technical networksAlignment of System Type and Canonical Shape• Assemblies seem like trees, or trees decorated with loose clusters at different hierarchical levels• Subway systems are like clusters with interior nodes and exterior pendants• Commuter rail lines can be trees, trees with cross-links, or grids– Grids arise when there is a robust intercity rail system and commuter trains can share these tracks– Also helps to have relatively flat ground– Trains must follow flat ground or cost a lot for tunnels– Flat ground is associated with water courses, as are locations of towns that need train service8/24/2006 © Daniel E Whitney 1997-2006 11/49Technical networksNetwork Models of Technical Systems• Need to carefully define what is a node and what is a link•Examples:– Assemblies: node = part; link = joint between two parts tat constrains at least one degree of freedom– Rail lines: node = rail junction or place where people can change train lines; link = rail– Electric circuit: node = circuit element (R, C, IC); link = wire– Distribution infrastructure: node = branch point, load, or sink; link = conductor ( pipe, wire)– Food web: node = species; link (directed) what eats what (can include cannibalism)8/24/2006 © Daniel E Whitney 1997-2006 12/49Technical networksDesign of Distribution Systems• Fundamental need is to “fill space” in some sense• Scaling issues• Cost per unit of capability or capacity• Levels of service: speed, choice of destination, equity• Context, legacy– Ability to run commuter trains on inter-city tracks– Ability to exceed service of legacy
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